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THE INSTITUTE FOR SYSTEMS RESEARCH

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www.isr.umd.edu

New Results on Modal Participation Factors:Revealing a Previously Unknown Dichotomy

Wael A. Hashlamoun, Munther A. Hassouneh, Eyad H. Abed

ISR TECHNICAL REPORT 20087


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New Results on Modal Participation Factors:

Revealing a Previously Unknown DichotomyWael A. Hashlamoun, Munther A. Hassouneh and Eyad H. Abed

AbstractThis paper presents a new fundamental approachto modal participation analysis of linear time-invariant systems,leading to new insights and new formulas for modal participationfactors. Modal participation factors were introduced over a quar-ter century ago as a way of measuring the relative participation ofmodes in states, and of states in modes, for linear time-invariantsystems. Participation factors have proved their usefulness in thefield of electric power systems and in other applications. However,in the current understanding, it is routinely taken for grantedthat the measure of participation of modes in states is identicalto that for participation ofstates in modes. Here, a new analysisusing averaging over an uncertain set of system initial conditions

yields the conclusion that these quantities (participation of modesin states and participation of states in modes) should not beviewed as interchangeable. In fact, it is proposed that a newdefinition and calculation replace the existing ones forstate in mode

participation factors, while the previously existing participationfactors definition and formula should be retained but viewedonly in the sense of mode in state participation factors. Severalexamples are used to illustrate the issues addressed and theresults obtained.

Index TermsParticipation factors, modal participation fac-tors, modal analysis, linear systems, stability, control systems.

I. INTRODUCTION

This paper presents new concepts, results, and formulasin the subject of modal participation analysis of linear time-

invariant systems. This topic is an important component of

the Selective Modal Analysis (SMA) framework introduced

by Perez-Arriaga, Verghese and Schweppe [7], [12] in the

early 1980s. A main construct in SMA is the concept of

modal participation factors (or simply participation factors).

Participation factors are scalars intended to measure the rel-

ative contribution of system modes to system states, and of

system states to system modes, for linear systems. The work of

these authors has had a major impact especially in applications

to electric power systems, where participation factors as they

were originally introduced have become a routine tool for the

practitioner and researcher alike.Since their introduction, participation factors have been

employed widely in electric power systems and other ap-

plications. They have been used for stability analysis, order

W. A. Hashlamoun was visiting the Institute for Systems Research, Univer-sity of Maryland, College Park, MD 20742 USA. He is now at the Departmentof Electrical and Computer Engineering, Birzeit University, Birzeit, Palestine(email: [email protected]).

M. A. Hassouneh is with the Institute for Systems Research and theDepartment of Mechanical Engineering, University of Maryland, CollegePark, MD 20742 USA (email: [email protected]).

E. H. Abed is with the Department of Electrical and Computer Engineeringand the Institute for Systems Research, University of Maryland, College Park,MD 20742 USA (email: [email protected]).

reduction, sensor and actuator placement, and coherency and

clustering studies (e.g., [7], [12], [8], [2], [5], [3], [9]). Several

researchers have also considered alternate ways of viewing

modal participation factors (e.g., [11], [4], [10]).

We study linear time-invariant continuous-time systems

x = Ax(t) (1)

where x Rn and A is a real n n matrix. We make theblanket assumption that A has a set of n distinct eigenvalues

(1,2, . . . ,n). The solution of (1) then takes the form of a

sum of modal components:

x(t) =n

i=1

eitci (2)

where the ci are constant vectors determined by the initial

conditionx0 and by the right and left eigenvectors ofA.

In their study of modal participation for the system (1), the

authors of [7], [12] selected particular initial conditions and

introduced definitions motivated by the calculation of relative

state and mode contributions using those initial conditions.

In this paper, we take a different approach, building on our

previous work [1], in which definitions of modal participation

factors are formulated by averaging relative contributions of

modes in states and states in modes over an uncertain set ofinitial conditions. In this approach, we consider initial condi-

tions to be unknown, and we take the view that performing

some sort of average over all possible initial conditions should

give a more reliable result than focusing attention on one

particular possible initial condition. The uncertainty in initial

condition can be taken as set-theoretic (unknown but bounded)

or probabilistic. We took the same basic approach in our

paper [1], but later found a subtle error in the calculation in

that paper for the case of state-in-mode participation factors.

Upon realizing this subtle error, we embarked on the present

research, in which we find a previously unnoticed dichotomy

in the two basic types of modal participation factors.

The main contribution of this paper is to reveal this pre-viously unknown dichotomy in modal participation analysis.

To wit, although the definitions obtained in [7], [12], and

which have been in wide use since their introduction, give

identical values for measures of participation of modes in

states and for participation of states in modes, these are

in fact better viewed as fundamentally different, and should

be calculated using two distinct formulas. Summarizing, the

main contribution of this paper is as follows: we propose

replacing the existing definition of participation factors with

two separate definitions that yield distinct numerical values

for participation of modes in states and for participation of


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states in modes. In this paper, the currently used participa-

tion factors measuring participation of states in modes are

replaced with a new first-principles definition, a particular

instance of which is an explicit formula given in Section

V. In addition, we show that our formula for participation

factors measuring participation of modes in states agrees

with the commonly used participation factors formula under

reasonable assumptions on the allowed uncertainty in the

system initial conditions. Thus, a dichotomy is proposed in

the calculation of participation factors.

The paper proceeds as follows. In Section II, the original

definitions of modal participation factors are recalled from [7],

[12]. In Section III, basic examples are used to illustrate

the need for an approach that yields distinct formulas for

measuring the two main types of modal participation: par-

ticipation of modes in states and participation of states in

modes. In Section IV, the approach we introduced to this

topic in [1] is recalled and discussed in light of the objectives

of the present paper. The discussion makes clear that this

approach, based on defining modal participation measures by

averaging over an uncertain set of initial conditions, readilyyields the original definition for participation of modes in

states, but does not easily yield a simple closed-form expres-

sion for measuring participation of states in modes. Next, in

Section V, a candidate closed-form formula is obtained for

modal participation factors that measure the participation of

states in modes; this is achieved by careful evaluation of the

general averaging formula from Section IV under a simplifying

assumption on the initial condition uncertainty. It is important

to note that to obtain this simple formula, a specific form is

assumed for the uncertainty in the system initial condition;

other assumptions on the initial condition uncertainty would

not lead to the same formula or any readily useable expression.

The derived formula is proposed since it reflects the effect ofinitial condition uncertainty and can be derived analytically. In

Section VI, a mechanical system example is used to illustrate

the usefulness of the explicit formula for state in mode par-

ticipation factors. In Section VII, an additional result is given

that relates only to mode in state participation factors. This

result expands the initial condition uncertainty assumptions

under which the traditional participation factors formula can

be shown to accurately measure mode in state participation

using the averaging formulation. Concluding remarks and

suggestions for future work are collected in Section VIII.

I I . ORIGINALD EFINITIONS OFM ODAL PARTICIPATION

FACTORS

In this section, the original definitions of modal participa-

tion factors are recalled from [7], [12]. Consider the linear

system (1), repeated here for convenience:

x = Ax(t) (3)

where x R n, and A is a real n n matrix. The authorsof [7], [12] also make the blanket assumption that A has

n distinct eigenvalues (1,2, . . . ,n). Let (r1

, r2, . . . , rn) beright eigenvectors of the matrix A associated with the eigen-

values (1,2, . . . ,n), respectively. Let (l1

, l2, . . . , ln) denote

left (row) eigenvectors of the matrix A associated with the

eigenvalues (1,2, . . . ,n), respectively. The right and lefteigenvectors are taken to satisfy the normalization [6]

lirj = i j (4)

where i j is the Kronecker delta:

i j

= 1 i= j0 i = jThe solution to (3) starting from an initial condition x(0) =x0

is

x(t) = eAtx0 (5)

Since the eigenvalues of A are distinct, A is similar to a

diagonal matrix. Using this, (5) can be rewritten in the form

x(t) =n

i=1

(lix0)eitri. (6)

From (6), xk(t) is given by

xk(t) =

n

i=1(lix0)e

itrik. (7)

A. Relative participation of the i-th mode in the k-th state

To determine the relative participation of the i-th mode

in the k-th state, the authors of [7], [12] select an initial

condition x0 =ek, the unit vector along the k-th coordinateaxis. As seen next, this choice is convenient in that it results

in a simple formula for mode-in-state participation factors. We

note that the derived formula for mode-in-state participation

factors agrees with that obtained using an uncertain initial

condition under general assumptions, as demonstrated in [1]

and in Section IV below. With this choice ofx0, the evolution

of the k-th state becomes

xk(t) =n

i=1

(likrik)e

it

=:n

i=1

pkieit

. (8)

The quantities

pki:=likr

ik (9)

are found to be unit-independent, and are taken in [7], [12] as

measures of the relative participation of the i-th mode in the

k-th state; pki is defined in [7], [12] as the participation factor

for the i-th mode in the k-th state.

B. Relative participation of the k-th state in the i-th mode

The relative participation of the k-th state in the i-th mode

is studied in [7], [12] by first applying the similarity transfor-

mation

z:= V1x (10)

to system (3), where V is the matrix of right eigenvectors of

A:

V= [r1 r2 rn] (11)


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and V1 is the matrix of left eigenvectors ofA:

V1 =

l1

l2

...

ln

. (12)

Then z obeys the dynamics

z(t) = V1AV z(t)

= z(t), (13)

where :=diag(1,2, . . . ,n), with initial condition z0 :=

V1x0. This implies that the evolution of the new state vector

components zi, i=1, . . . , n is given by

zi(t) = z0ie

it

= lix0eit

=

n

k=1

(likx0k)

eit. (14)

For a real eigenvalue i, clearly z i(t) represents the evolutionof the associated mode. Ifi is not real, then the associated

mode is sometimes taken to be zi(t), but can also be taken asthe combination ofzi(t)and its complex conjugate z

i(t), which

reflects the influence of the eigenvalue i . In the latter ap-

proach, we view i and i as representing the same complex

frequency. In the past, the former convention was used in most

publications. In this paper, we allow both interpretations, but

we will find it convenient to use the latter point of view when

deriving a new state-in-mode participation factors formula for

the case of complex eigenvalues.

In order to determine the relative participation of the k-

th state in the i-th mode, the authors of [7], [12] select aninitial condition x0 =ri, the right eigenvector associated withi. As seen next, this choice is convenient in that it results in

a simple formula for state-in-mode participation factors. We

will revisit this later using an uncertain initial condition, and

obtain a different result. With this choice of initial condition,

the evolution of the i-th mode becomes

zi(t) = lirieit

=

n

k=1

likrik

eit

= n

k=1pkieit. (15)

Based on (15), the authors of [7], [12] propose the formula

pki=likr

ik (16)

as a measure of the relative participation of the k-th state in

the i-th mode.

Note that (9), (16) provide identical formulas for participa-

tion of modes in states and participation of states in modes,

respectively. For this reason, the same notation pki was used

for both types of participation factors until now.

III. MOTIVATINGE XAMPLES S HOWINGI NADEQUACY OF

PARTICIPATIONFACTORS F ORMULA AS AM EASURE OF

STATE INM OD E PARTICIPATION

In this section, by way of motivation for the subsequent

analysis, two examples are given that show the need for a new

definition and a new formula for state in mode participation

factors.

Example 1 Consider the two-dimensional system

x1

x2

=

a b

0 d

A

x1x2

where a, b and dare constants with a =d. The eigenvaluesofAare 1=a and 2=d. The right eigenvectors associatedwith 1 and 2 are

r1 =

1

0

and r2 =

1da

b

,

respectively. The left eigenvectors associated with 1 and 2and satisfying the normalization (4) are

l1 =

1 bad

and l2 =

0 b

ad

,

respectively.

Before calculating the participation factors measuring the

influence of statesx1and x2in mode 1, we write the evolution

of mode 1 explicitly. Using (14), we have

z1(t) = l1x0e1t

=

1

b

ad

x01x02

e1t

=

x01+

b

adx02

e1t. (17)

Note that the evolution of mode 1 is influenced by both x01and x02, with the relative degree of influence depending on the

values of the system parameters a, b and d.

Calculating the participation factors using the original def-

inition as recalled in the foregoing section, we find the

participation factor for state x1 in mode 1 is p11= l11 r

11= 1,

while the participation factor for state x2 in mode 1 isp21= l

12 r

12= 0. Thus, the original definition of participation

factors for state in mode participation indicates that state x2has much smaller (even zero) influence on mode 1 compared

to the influence coming from state x1, regardless of the values

of system parameters a , b and d. This is in stark contradiction

to what we observed using the explicit formula (17), and begs

for a re-examination of the basic formula for state-in-mode

participation factors.

For simplicity, we use the terminology mode i in place of the modeassociated with eigenvalue i.


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Example 2 Consider the two-dimensional system

x1

x2

=

1 1

d d

A

x1x2

where d= 1 is a constant. The eigenvalues of A are 1= 0

and 2= 1d. The right eigenvectors associated with 1 and2 are

r1 =

1

1

and r2 =

1

d

,

respectively. The left eigenvectors associated with 1 and 2and satisfying the normalization (4) are

l1 = d

1d1

1d

and l2 =

11d

11d

,

respectively. Denote by Vthe matrix of right eigenvectors of

A:

V= [r1 r2] = 1 11 d .From the normalization condition (4), we can immediately

write

V1 =

l1

l2

=

d1d

11d

11d

11d

.

The evolution of the modes can be obtained using the

diagonalizing transformation z:= V1x as was done in (10)-(13). The system modes are found to be

z1(t)z2(t)

=

l1x0e1t

l2x0e2t

= d1dx01 11dx02e1t11d

x01+x

02

e2t

. (18)

Based on the original definition of participation factors, the

participation factor for state x1 in mode 2 is p12= r21l

21=

11d,

and the participation factor for state x2 in mode 2 is p22=r22l

22=

d1d. Clearly, in general p12 =p22. However, from (18)

we have the equation

z2(t) = 1

1d

x01+x

02

e2t (19)

for the second mode z2(t), from which we observe that statex1 and state x2 participate equally in mode 2 since z2(t)

depends on the initial condition x

0

through the sum x

0

1+x0

2.Again, we find that the state-in-mode participation factors as

commonly calculated yield conclusions that are very much

at odds with what one might consider reasonable based on

explicit calculation of the evolution of system modes as they

depend on initial conditions of the state variables.

The inadequacy of the original state-in-mode participation

factors formula has been demonstrated in the two examples

above. This motivates the need for a new formula that better

assesses the influence of system states on system modes.

IV. INITIAL C ONDITION U NCERTAINTY A PPROACH:

APPLICATION TOD ERIVATION OFM OD E-IN-S TATE

PARTICIPATIONFACTORS

For systems operating near equilibrium, it is often reason-

able to view the system initial condition as being an uncertain

vector in the vicinity of the system equilibrium point. In this

paper, and in the authors previous work [1], we approachthe problem of measuring modal participation by averaging

relative contributions over an uncertain set of initial conditions.

In this section, we summarize this approach as it applies to

the definition and calculation of mode-in-state participation

factors. We carry the approach through to its conclusion

for this problem, obtaining an explicit formula for mode-

in-state participation factors. As mentioned above, the final

formula we obtain using this approach agrees in this case

with the previously existing expression pki. However, in the

next section, such a happy coincidence will not occur for the

more delicate situation of defining and calculating state-in-

mode participation factors.

Next, we recall from our previous work [1] a basic definition

of relative participation of a mode in a state. This definition

involves taking an average over system initial conditions of a

measure of the relative influence of a particular system mode

on a system state. The initial condition uncertainty can be

taken as set-theoretic or probabilistic. In the set-theoretic for-

mulation, the participation factor measuring relative influence

of the mode associated with i on state xkcan be defined as

pki := avgx0

S

(lix0)rikx0k

(20)

whenever this quantity exists (however, see Remark 1 below

for another possible definition for the case of complex i).

Here, x0k=ni=1(l

ix0)rik is the value of xk(t) at t= 0, andavgx0S is an operator that computes the average of a

function over a set S Rn (representing the set of possiblevalues of the initial condition x0). We assume that the initial

condition uncertainty set S is symmetric with respect to each

of the hyperplanes{xk=0}, k=1, . . . , n.

In the definition in [1] that starts with a probabilistic

description of the uncertainty in the initial condition x0, the

average in (20) is replaced by a mathematical expectation.The general formula for the participation factor pki measuring

participation of mode i in state xkbecomes

pki := E

(lix0)rik

x0k

(21)

where the expectation is evaluated using some assumed joint

probability density function f(x0) for the initial conditionuncertainty (of course, this definition applies only when the

expectation exists).


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Expanding the inner product term in (21), we find

pki = E

n

j=1

(lijx0j )r

ik

x0k

= E

(likx

0k)r

ik

x0k

+E

n

j=1, j=k

(lijx0j )r

ik

x0k

= likrik+

n

j=1, j=k

lijrikEx0j

x0k

. (22)

The second term in (22) vanishes when the components of

the initial condition vector x01,x02, . . . ,x

0n are independent with

zero mean [1]. Therefore, under the assumption that the initial

condition components x01,x02, . . . ,x

0n are independent with zero

mean, the participation of the i-th mode in the k-th state is

given by the same expression originally introduced by Perez-

Arriaga, Verghese and Schweppe [7], [12]:

pki = likr

ik. (23)

This result can also be obtained using the set-theoretic aver-

aging formula (20) [1].Remark 1: (Alternate Definition of Mode-in-State Partici-

pation Factor for a Complex Mode) For a complex eigen-

value i, the associated mode is taken above as the term

containing eit in the system response (2). However, we can

alternately view this mode as consisting of the combined

contributions fromiand its complex conjugate eigenvalue i.

This viewpoint is easily seen to lead, under the same symmetry

hypotheses as above, to the following alternate expression for

the participation factor of the mode associated with i and i

in state xk:

pki = 2Re {likr

ik}. (24)

V. NEW D EFINITION OFPARTICIPATIONFACTORS

MEASURING PARTICIPATION OFS TATES INM ODES

In this section, a new definition and calculation are given

for participation factors measuring contribution of states in

modes. The probabilistic approach presented in the previous

section is used, where the initial condition is assumed to satisfy

a joint probability density function. In order to obtain an

explicit formula from the new general definition of state-in-

mode participation factors, we find that it is necessary to make

an assumption on the probability distribution of the initial con-

dition which is more constraining than what was needed in the

analysis above for mode-in-state participation factors. Thus,

the explicit formula derived in this section should be viewed inthe pragmatic sense that it provides an easy to use expression

that reflects initial condition uncertainty. Other assumed forms

of uncertainty may not lead to explicit formulas, although a

formula requiring numerical evaluation of integrals can always

be obtained from the definition. The explicit formula obtained

here differs from the single formula (16) that is currently

used to measure both state-in-mode participation and mode-

in-state participation, while the currently used formula (16)

is retained here as a measure of mode-in-state participation

(noting that the alternate formula (24) can also be used for

the case of a complex mode). This dichotomy represents a

significant departure from current practice. We will also use

the new formula to revisit the examples of Section III.

Consider the general linear time-invariant continuous-time

system given in (3), repeated here for convenience:

x = Ax(t) (25)

The evolution ofz i(t), i =1, . . . , n, was obtained in Section II

as

zi(t) = eitlix0

= eitn

j=1

(lijx0j ). (26)

This equation shows the contribution of each component x0j ,

j = 1, . . . , nof the initial state x0 tozi(t). Recall also that for thecase of a real eigenvalue i, z i(t) is identically the i-th mode,while, for a complex eigenvalue i, the associated mode can be

taken as zi(t) or as the combination ofzi(t) and its conjugate:zi(t)+z

i(t) = 2Re{zi(t)}. The following general definition of

state-in-mode participation factors is obtained by averaging the

relative contribution ofx0

k in the i-th mode and evaluating theresult at t=0. In this definition, we take the mode associatedwith a complex eigenvalue as 2Re {zi(t)}, i.e., the combinationof modal components due to the eigenvalue and its conjugate.

Had we decided to view the mode associated with a complex

eigenvalue i as zi(t) alone, we would use the first expressionin the definition below for both the case of a real and a

complex eigenvalue. However, the derivation following the

basic definition below of a simple final formula would become

unwieldy for the complex eigenvalue case.

Definition 1: For a linear time-invariant continuous-time

system (25), the participation factor for the k-th state in the

i-th mode is

ki :=

E

likx0

k

z0i

ifi is real

E

(lik+l

ik

)x0kz0i+z

0i

ifi is complex

(27)

whenever the expectation exists.

Note that in (27), the notation z0i means zi(t= 0) =lix0 and

the asterisk denotes complex conjugation. Also, analogous to

the approach in Section IV and the original work [7], [12], the

quantities being evaluated represent the contribution of state xkto a mode divided by the total mode evaluated at time t= 0(however, see the Conclusions section about the possibility

of measuring modal participation effects over time). Note

also that Definition 1 always yields real-valued participationfactors.

Unfortunately, even under an assumption such as symmetry

of the initial condition uncertainty, there is no single closed-

form expression for the state in mode participation factors

ki. To obtain a simple closed-form expression for the state

in mode participation factors ki using (27), we need to

find an assumption on the probability density function f(x0)governing the uncertainty in the initial condition x0 that allows

us to explicitly evaluate the integrals inherent in the definition.

In the remainder of this section, we assume that the units

of the state variables have been scaled to ensure that the


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probability density function f(x0) is such that the componentsx01,x

02, . . . ,x

0n are jointly uniformly distributed over the unit

sphere in Rn centered at the origin:

f(x0) =

k ||x0|| 10 otherwise

(28)

(This is the same as assuming a uniform distribution in

an ellipsoid that is centered at the origin and symmetricwith respect to the coordinate hyperplanes in the original

state variable units, a physically palatable assumption and

independent of units by construction.) The constant kis chosen

to ensure the normalization||x0 ||1

f(x0)dx0 =1. (29)

The value of the constant kcan be determined by evaluating

the integral in (29) using f(x0) given in (28):

||x0||1 f(x0)dx0 = ||x0||1 kdx

01dx

02 . . . dx

0n=kVn=1 (30)

whereVn is the volume of the unit sphere in Rn. The constant

k is then given by

k= 1

Vn. (31)

The following Lemma will be used below.

Lemma 1: For vectors a Cn, b Rn with b =0 we have

||x||1

aTx

bTxdnx =

aTb

bTbVn (32)

where dnx denotes the differential volume elementdx1dx2 dxn, and Vn is the volume of a unit sphere inRn which is given by

Vn=

2, n=1, n=22n

Vn2, n 3(33)

Proof: The proof below is for the case a, b Rn, b =0. Thecase a Cn and b Rn follows by linearity. Consider thetransformationx =Qy where the matrix Q is chosen to be anorthogonal matrix (i.e.,Q1 = QT) with first columnb1 := b||b||

(i.e., Q = [b1 Q1]). The remaining columns ofQ1 are chosento be orthogonal to b1, i.e.,

(b1)TQ1=0. (34)

With this transformation, since Q1 = QT, we have that||x||2 =xTx= (Qy)TQy=yTQTQy=yTy= ||y||2. Also, sinceQ is orthogonal, det(Q) =1 and dnx=dny.

Writing the vector y as

y=

y1

y

with y1 Rand yRn1, the integral in (32) can be expressed

as ||x||1

aTx

bTxdnx =

||y||1

aTQy

bTQydny

=||y||1

aT[b1 Q1]y

bT[b1 Q1]ydny

= ||y||1 aTb1y1+ a

TQ1y

bTb1y1+ bTQ1y

=0

dny. (35)

The expression (35) can be further simplified as follows:||x||1

aTx

bTxdnx =

aTb1

bTb1

||y||1

dny (36)

+ 1

bTb1

||y||1

aTQ1y

y1dn1y

dy1.

The first integral on the right of (37) evaluates to Vn, the

volume of a unit sphere in Rn, whereas the second integral

vanishes (it is improper but the Cauchy principal value is 0).

Therefore, ||x||1

aTx

bTxdnx =

aTb

bTbVn (37)

whereVn is given by (33). This completes the proof.

Next, the relative participation of the k-th state in the i-th

mode is evaluated using Definition 1 under the assumption

above on the distribution of the initial condition x0. Before

proceeding, we recall the relationship between x0 and z0:

x0 = V z0

=n

j=1

rjz0j . (38)

A. Participation in a mode associated with a real eigenvalue

To determine the participation of the k-th state in a real

mode associated with a real eigenvalue i, we substitute x0k=

nj=1 r

jkz

0j in (27):

ki = E

likx

0k

z0i

= E

lik

nj=1 r

jkz

0j

z0i

= E likr

ikz

0i

z0

i +n

j=1, j=i

likrjkE

z0j

z0

i = likr

ik +

n

j=1, j=i

likrjkE

z0j

z0i

. (39)

Note that the first term in (39) coincides with pki, the origi-

nal participation factors formula. We will find that, in general,

the second term in (39) does not vanish. This is true even

in case the components x01,x02, . . . ,x

0n representing the initial

conditions of the state are assumed to be independent. This is

due to the fact that the second term involves the components

of z0 (i.e., z01 ,z02, . . . ,z

0n) which need not be independent even

under the assumption that the x0kare independent, due to the


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7

transformationz0 = V1x0. This was overlooked in [1], leadingto the incorrect conclusion there that the second term in (39)

vanishes.

We now use Lemma 1 to simplify the expression (39) for

the participation factor for thek-th state in thei-th (real) mode:

ki = likr

ik +

n

j=1, j=i

likrjkE

z0j

z0i . (40)

Substitutingz0i =lix0 into (40) yields

ki = likr

ik +

n

j=1, j=i

likrjkE

ljx0

lix0

= likrik +

n

j=1, j=i

likrjkE

g(x0)

(41)

where g(x0) takes the form

g(x0) = l

j1x

01+ l

j2x

02+ + l

jnx

0n

li1x01+ l

i2x

02+ + l

inx

0n

. (42)

Denote a:= (lj1 , l

j2 , . . . , l

jn)T and b:= (li1, l

i2 , . . . , l

in)

T. The ex-

pected value of g(x0) is

E{g(x0)} =||x0||1

g(x0)f(x0) dx0

= k||x0||1

aTx0

bTx0 dx0 (43)

Using Lemma 1, which applies since b is real, and the

normalizationkVn=1 from (31), this integral reduces to

E{g(x0)} = aTb

bTbkVn

= aT

bbTb

. (44)

Substituting (44) into (41) yields a key result of this paper, a

new formula for the participation factor for state xk in a real

mode:

ki = likr

ik +

n

j=1, j=i

likrjk

lj(li)T

li(li)T . (45)

Remark 2: Under the initial condition uncertainty assump-

tion based on which (45) was obtained, the participation factor

for thei-th mode in thek-th state equals the participation factor

for the k-th state in the i-th mode (i.e., ki= pki) if the left

eigenvectors of the system matrix A are mutually orthogonal,i.e.,

lj(li)T =0, for j, i=1, 2, . . . , n, i = j.

This is a very restrictive case (which applies, for instance,

when the system matrix, being real, is symmetric).

Next, we derive the following expression equivalent to (45)

ki = (lik)

2

li(li)T

= (lik)

2

nj=1(l

ij)

2. (46)

This expression is easily obtained from Definition 1 and

Lemma 1 as follows. Recall Definition 1, the general

averaging-based definition for state-in-mode participation fac-

tors for the case of a real eigenvalue:

ki := E

likx

0k

z0i

. (47)

Substitutingz0

i =lix0 in (47) yields

ki = E

likx

0k

lix0

=E

like

kx0

lix0

. (48)

Denote

a := likek =l ik[0 . . . 0 1 0 . . . 0]

T,

b := (li)T.

Using Lemma 1 and the normalization kVn= 1, (48) reducesto

ki = aTb

bTb=

likek(li)T

li(li)T =

(lik)2

li(li)T , (49)

which is exactly (46).

B. Participation in a mode associated with a complex conju-

gate pair of eigenvalues

To determine the participation factor for a state in a complex

mode, i.e., a mode associated with a complex conjugate pair

of nonreal eigenvaluesi, i, we use the second case of (27):

ki = E

(lik+ l

i

k)x0k

z0i +z0

i

= E

Re{lik}x

0k

Re{z0i }

. (50)

Substitutingz0i =lix0 in (50) yields

ki = E

Re{lik}x

0k

Re{lix0}

=E

Re{lik}x

0k

Re{li}x0

. (51)

Equation (51) can be rewritten as

ki = E

Re{lik}(e

k)Tx0

Re{li}x0

= Re{lik} E

(ek)Tx0

Re{li}x0

. (52)

Next, we obtain formulas analogous to (45) and (46) above,

but now giving participation factors measuring participation of

states in a complex mode.First, to determine a formula analogous to (45), substitute

z0i = lix0 and x0k =

nj=1 r

jkz

0j =

nj=1 r

jkl

jx0 and apply and

Lemma 1 to obtain

ki = E

Re{lik}x

0k

Re{li}x0

= E

Re{lik}

nj=1 r

jkl

jx0

Re{li}x0

= Re{lik} E

nj=1 r

jkl

jx0

Re{li}x0

. (53)


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8

Let a=

nj=1 r

jkl

jT

and b= (Re{li})T. Since b is real,

we can invoke Lemma 1 and the normalization kVn = 1 toreduce (53) to

ki = Re{lik}

nj=1 r

jkl

j

(Re{li})T

Re{li}(Re{li})T . (54)

This formula can be rewritten as

ki= Re{lik}

rikl

i

(Re{li})T

Re{li}(Re{li})T+

n

j=1, j=i

Re{lik}rjkl

j(Re{li})T

Re{li}(Re{li})T .

(55)

which is the desired form analogous to (45). We observe that if

this formula is applied to a simple real eigenvalue i, implying

that the associated eigenvector can also be taken as real, the

formula indeed reduces to the formula (45) that was derived

for the case of a real mode. Thus, formula (55) provides a

general expression for state-in-mode participation factors for

systems without any restriction on the eigenvalues besides that

they are distinct.

Next, we obtain another equivalent formula for the ki for

a complex mode, but this time in a form analogous to the

expression (46) derived above for the case of real eigenvalues.

We use Lemma 1 to simplify the expression (52). Taking a=ek andb = (Re{li})T in Lemma 1 and using the normalizationkVn=1, (52) reduces to

ki = Re{lik}

(ek)T(Re{li})T

Re{li}(Re{li})T

= Re{lik} Re{lik}

Re{li}(Re{li})T .

Finally,

ki= (Re{lik})

2

Re{li}(Re{li})T . (56)

This expression is an alternate form of the newly proposed

formula (55) for participation factors measuring participation

of a state xk in a mode. Both expressions apply for a mode

associated with a complex conjugate pair of eigenvalues iandi , as well as for a simple real eigenvalue i.

Remark 3: Although care was taken with respect to se-

lecting units in deriving the formulas for state-in-mode par-

ticipation factors, the final formulas are not themselves in-

dependent of units. The independence with respect to statevariable units that occurs in the definitions of mode-in-state

participation factors is a fortunate coincidence for certain

choices of initial conditions or under certain initial condition

symmetry assumptions. However, no such coincidence occurs

in the quantification of state-in-mode participation. One way

of viewing this is as follows. Units are important, in the sense

that they should be chosen so that a unit variation in the initial

condition of any state variable has a similar likelihood as a unit

variation in the initial condition of any other state variable of

the system. That is the spirit of the assumption made above

that the distribution of initial conditions of the state vector can

be mapped by changes of units to a uniform distribution on a

unit sphere in Rn.

Next, we revisit Examples 1 and 2 using the newly derived

formula for state in mode participation factors, and compare

the results to the participation factors obtained using the

original definitions. Note that all eigenvalues in these examples

are real, so the formula (45) applies as a (new) measure of

state-in-mode participation factors ki (as does the equivalent

formula (46)).

Example 1 Revisited

For Example 1, the participation factors for states x1 and x2in mode 1 based on the new formula (45) are

11 = (ad)2

(ad)2 + b2 and 21=

b2

(ad)2 + b2,

respectively. The participation factors for states x1 and x2 in

mode 1 based on the original formula are p11= 1 and p21= 0,respectively.

As we observed previously in our discussion of Example1, the original formula for participation factors erroneously

indicates that the participation of state x2 in mode 1 is

zero. The coupling between state x2 and state x1 in the

system dynamics is not reflected in the original formula for

participation factors (the pki), whereas this coupling between

state variables is reflected in the result of applying the new

formula (for the ki).

Example 2 Revisited

For Example 2, the participation factors for states x1 and x2in mode 2 based on the new formula (45) are

12= 12

and 22= 12

,

respectively. The participation factors for states x1 and x2 in

mode 2 based on the original formula are

p12= 1

1dand p22=

d

1d,

respectively.

The results using the new formula more faithfully reflect the

relative contributions of the initial conditions of the two state

variables to the evolution of mode 2, which is given explicitly

by the formula

z2(t) =

1

1dx01+x02e2t, (57)Here it is clear that z2(t) is equally influenced by x

01 and x

02

since it depends on the initial condition x0 through the sum

x01+x02.

VI. A NUMERICALE XAMPLE: A TWO-M AS S

MECHANICALSYSTEM

Consider the translational mechanical system depicted in Fig-

ure 1, where y1(t) and y2(t)denote the displacements of mass1 and mass 2, respectively, from the static equilibrium [13].

The system parameters are the masses m1 and m2, viscous


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9

damping coefficients c1 and c2, and the spring constants k1and k2.

A state space representation is obtained by defining the

system states as

x1(t) = y1(t)

x2(t) = y1= x1

x3(t) = y2(t)x4(t) = y2= x3.

The system dynamics is described by the linear time-invariant

differential equation [13]

x = Ax

where

A =

0 1 0 0

k1+k2m1

c1+c2m1

k2m1

c2m1

0 0 0 1k2m2

c2m2

k2m2

c2m2

.

With the system parameters selected as m1=39 kg, m2=17kg, c1= 19 Ns/m, c2=33 Ns/m, k1= 374 N/m, and k2=196N/m, the system state dynamics matrix A becomes

A =

0 1 0 0

14.6154 1.3333 5.0256 0.84620 0 0 1

11.5294 1.9412 11.5294 1.9412

.

Fig. 1. Mechanical system.

The eigenvalues ofA are 1,2=0.217j2.315 and3,4=1.4203 j4.2935. We denote the modes associated withthe eigenvalues as z1(t) and z2(t), respectively. The dominantmode is the one associated with the complex conjugate pair

of eigenvalues closest to the imaginary axis, i.e., 1,2. Modes

associated with eigenvalues close to the imaginary axis are of

considerable interest as they can be used as an indication of

closeness to system instability. Therefore, our emphasis in this

example will be on z1(t).The right (column) eigenvectors of the system matrix A

associated with 1 and 3 are

r1 =

0.0124j0.19380.4461 +j0.07080.0321j0.3425

0.7999

and (58)

r3 =

0.0716 +j0.10210.5402 +j0.16240.0553j0.1673

0.7970

,

respectively. Note that since 2= 1 and 4=

3, we have

thatr2 =r1 and r4 =r3, where an asterisk denotes complexconjugation. The left (row) eigenvectors of the system matrix

A associated with 1 and 3 are

l1 =

0.3122 +j1.2059

0.4806 + j0.05390.2760 + j0.7884

0.

3730 j0.

0171

T

,

l3 =

0.3134 j2.42510.4824j0.17760.2771 +j1.3357

0.2530 + j0.1903

T

(59)

respectively, and l2 =l1; l4 =l3.The magnitudes of the original participation factors pki

evaluated using (23) for this example are given in Table I.

The state in mode participation factors ki evaluated using the

new formula (56) are given in Table II.

TABLE I

PARTICIPATION FACTORS,pki , BASED ON ORIGINAL FORMULA .

mode 1 mode 2

x1 |p11| =0.2420 |p12 | =0.3050

x2 |p21| =0.2184 |p22 | =0.2900

x3 |p31| =0.2874 |p32 | =0.2404

x4 |p41| =0.2987 |p42 | =0.2523

TABLE II

STATE IN MODE PARTICIPATION FACTORS , ki , BASED ON NEW FORMULA.

mode 1 mode 2

x1 11=0.1792 12=0.2082

x2 21=0.4248 22=0.4934

x3 31=0.1401 32=0.1628

x4 41=0.2558 42=0.1357

We observe that the participation factors given in Table I,

which are calculated using the original definition of partic-

ipation factors, differ from the state in mode participation

factors in Table II calculated using the new formula. For

instance, according to Table I, the state that participates most

in mode 1 is x4, whereas according to Table II, the state

that participates most in mode 1 is x2. To demonstrate that

state x2 participates more than other state variables in mode

1, we calculate the evolution of mode 1 (z1(t)) due to different

settings in the initial conditions. Specifically,z1(t)is calculatedfor the following set of initial conditions: x0 = [0.1, 0, 0, 0]T,x0 = [0, 0.1, 0, 0]T, x0 = [0, 0, 0.1, 0]T, and x0 = [0, 0, 0, 0.1]T.We select the initial conditions in this way (0 in all but one of

the state variables) to distinguish the influence of each of the

state variables on mode 1. The simulation results are depicted

in Figure 2, which shows plots of Re{z1(t)} for the variousinitial conditions. Figure 2 shows that the initial condition

component x02 gives the largest effect on mode 1 at t= 0compared to all other state variables. This agrees with what is

predicted using the new formula for state in mode participation

factors (see Table II). In other words, mode 1 can be excited


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10

by an initial condition on any of the state variables, however,

the highest excitation at t= 0 comes from x02 and the leastexcitation comes from x03.

The state in mode participation factors calculated based on

the new formula derived in this paper can be used to determine

the relative degrees by which system states excite a particular

mode in the system. This can be useful in stability monitoring

applications. For example, in stressed electric power systems,

typically there is one critical mode that needs to be monitored.

If the system is operating exactly at an equilibrium point and

not influenced by disturbances, this mode cannot be observed

in the outputs of the system. In order to monitor the critical

mode and detect closeness to instability, a small perturbation

signal is applied to the system and the response is measured.

The state in mode participation factors can help in selecting

a location for applying the perturbation signal in order to

achieve the highest excitation in the critical mode to achieve

the clearest possible indication on how close the system is to

instability.

0 1 2 3 4 5 6 7 8 9 100.15

0.1

0.05

0

0.05

0.1

time [sec]

Rez1

(t)

change in x0

1

change in x02

change in x0

3

change in x0

4

Fig. 2. The effect on mode 1 of a perturbation of 0.1 away from equilibriumin the initial condition component x01 (solid), x

02 (dotted), x

03 (dash-dot) and

x04 (dashed).

VII. A FURTHER R EMARK ONM ODE INS TATE

PARTICIPATIONFACTORS

In this section, an additional result is given that relates

only to mode in state participation factors. The result is that,

in the averaging formulation, an additional possible set of

initial condition uncertainty assumption is found to also lead to

the traditional participation factors formula for mode in state

participation factors.

In Section IV, we showed that when the components of theinitial condition vector x01;x

02, . . . ,x

0n, are independent random

variables with zero mean, the participation of mode i in state

xk is given by

pki = likr

ik. (60)

In this section, we show that this expression remains valid if

the components of the initial condition x0, x0j , j=1, 2, . . . , n,are assumed to be jointly uniformly distributed over the unit

sphere in Rn (see (28) for the expression of the probability

density function in this case). Note that under this assumption,

the random variables x0j are no longer independent.

Consider the general expression (22) for the mode in state

participation factor pki, repeated here for convenience:

pki = likr

ik+

n

j=1, j=k

lijrikE

x0j

x0k

. (61)

A typical term in the summation on the right side of (61) takes

the form

Ex0j

x0k

j=k

=||x0 ||1

x0j

x0kf(x0) dx0

= k||x0||1

x0j

x0kdx0 (62)

Denote

a := (0, . . . , 0, 1jth

, 0, . . . , 0)T,

b := (0, . . . , 0, 1

kth

, 0, . . . , 0)T.

The integral in (62) can be expressed as

E

x0j

x0k

j=k

= k

||x0 ||1

aTx0

bTx0 dx0 (63)

Using Lemma 1, this expression reduces to

E

x0j

x0k

j=k

= aTb

bTb=0. (64)

Therefore, the second term in (61) vanishes and, under the

new assumptions, the mode in state participation factors are

still given by

pki = li

kri

k (65)

We can therefore conclude that (65) is a valid formula for

mode in state participation factors under any of the following

assumptions on the initial conditions:

1) The initial conditionx0 is taken to lie in an uncertainty

set S which is symmetric with respect to each of the

hyperplanes x0k= 0, k=1, 2, . . . , n.2) The initial condition components are independent ran-

dom variables with marginal density functions which are

symmetric with respect to x0k= 0, k=1, 2, . . . , n3) The initial condition components,x0j , j=1, 2, . . . , n, are

jointly uniformly distributed over a sphere centered at

the origin.

VIII. CONCLUSIONS

We have presented a new fundamental approach to quan-

tifying the participation of states in modes for linear time-

invariant systems. We have proposed that a new definition

and formula replace the commonly used participation factors

formula for measuring participation of states in modes, while

recommending the previously used participation factors for-

mula be retained as a quantification of participation of modes

in states. The analysis presented in this paper uses averaging

over an uncertain set of system initial conditions. The analysis


12/12

11

led to the conclusion that participation of modes in states

and participation of states in modes should not be viewed

as equivalent or interchangeable. Examples were given to

demonstrate that the original formula for participation factors

is not convincing as a measure of state in mode participation.

Moreover, these examples demonstrated the applicability and

usefulness of the new formula for state in mode participation

factors.

It is interesting that while the problem addressed in this

paper relates to a very simple and well studied class of

systems, considerable effort was required to revisit what may

seem to be a basic issue, namely modal participation. Indeed,

it appears that work is needed on further related matters, such

as implications of the results for sensor and actuator place-

ment, for system monitoring to detect impending instability,

for order reduction, and possibly for coherency studies of

power networks, among other issues. Also, from the results

in Figure 2 for the mechanical example, the relative sizes

of modal participations at the initial time instant might differ

from those over a time interval. For some applications, it may

be desirable to have analytical measures of modal participationthat quantify modal participation over time.

ACKNOWLEDGMENTS

The authors are grateful to Profs. Anan M.A. Hamdan,

Mohamed S. Saad, Andre L. Tits, Dr. David Lindsay, and

Mr. Li Sheng for helpful remarks received in the course of

this work. The support of the Fulbright Scholars Program, the

National Science Foundation and the Office of Naval Research

are gratefully acknowledged.

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[2] L.C.P. da Silva, Y. Wang, V.F. da Costa, and W. Xu, Assessmentof generator impact on system power transfer capability using modalparticipation factors,IEE Proceedings on Generation, Transmission and

Distribution, Vol. 149, No. 5, pp. 564570, Sept., 2002.[3] I. Genc, H. Schattler, and J. Zaborszky, Clustering the bulk power

system with applications towards Hopf bifurcation related oscillatoryinstability, Electric Power Components and Systems, Vol. 33, No. 2,pp. 181198, Feb, 2005.

[4] A.M.A. Hamdan, Coupling measures between modes and state vari-ables in power-system dynamics,International Journal of Control, Vol.43, No. 3, pp. 10291041, 1986.

[5] Y.Y. Hsu and C.L. Chen, Identification of optimum location of stabilizerapplications using participation factors, IEE Proceedings-C GenerationTransmission and Distribution, Vol. 134, No. 3, pp. 238244, May, 1987.

[6] P. Lancaster and M. Tismenetsky, The Theory of Matrices. New York:

Academic Press, 1985.[7] I.J. Perez-Arriaga, G.C. Verghese, and F.C. Schweppe, Selective modal

analysis with applications to electric power systems, Part I: Heuristicintroduction,IEEE Transactions on Power Apparatus and Systems , Vol.101, No. 9, pp. 31173125, Sept. 1982.

[8] I.J. Perez-Arriaga, G.C. Verghese, F.L. Pagola, J.L. Sancha and F.C.Schweppe, Developments in selective modal analysis of small-signalstability in electric power systems, Automatica, Vol. 26, No. 2, pp.215231, 1990.

[9] G. Rogers, Power System Oscillations. Boston: Springer, 1999.[10] M. Tarokh, Measures for controllability and eigenvalue mobility, in

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[11] J.E. Van Ness and F.D. Dean, Interaction between subsystems of apower system, in Proc. American Control Conference, Baltimore, June1994, pp. 1553-1557.

[12] G.C. Verghese, I.J. Perez-Arriaga, and F.C. Schweppe, Selective modalanalysis with applications to electric power systems, Part II: Thedynamic stability problem,IEEE Transactions on Power Apparatus andSystems, Vol. 101, No. 9, pp. 31263134, Sept. 1982.

[13] R.L. Williams II and D.A. Lawrence, Linear State-Space ControlSystems. Hoboken, NJ: John Wiley & Sons, 2007.

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